In figure REF (c), we compare the drag reduction results from all the 3D numerical simulations of [1]} with the leading-order drag reduction predictions from our 1D model.
These simulations span the whole parameter space characteristic of realistic microchannel applications.
In order to approximate the drag reduction from the streamwise slip length results provided by [1]}, we integrate the streamwise velocity field when \(\phi _z=0\)  for the solid-walled and SHS flows.
For the flow over a single SHS, we replace the mixed boundary conditions at \(y=0\)  with \(\lambda _e u_{0y} - u_0 = 0\) , to find \(u_0\)  as a function of \(y\)  and \(\lambda _e\) .
We then use \({DR}_0 = (\Delta p_R - \Delta p_0) / (\Delta p_R - \Delta p_U)\)  to relate the leading-order drag reduction and streamwise slip length, \({DR}_0 = 1 - 1 / (2 \lambda _e + 1)\) , in a similar manner to [3]}.
There is no obvious correlation between the scatter and the size of \(\epsilon \)  (shown in colour), which varies up to \(\epsilon \approx 0.5\)  as highlighted by the colorbar in figure REF (c).
The majority of the data points have \({DR}_0 < 0.5\)  and lie in region M.
Those that have \({DR}_0 > 0.5\)  are classified as belonging to region A when \(\alpha < 1\)  or D when \(\alpha > 1\) , as highlighted by the colorbar in figure REF (d).
The scatter could be due to unresolved cross-channel concentration gradients or streamwise boundary layers at the ends of the plastron; both hypotheses require further testing against targeted numerical simulations.
Nonetheless, our theory compares well with simulations for \({DR}_0 > 0.1\)  in figure REF (d), where the root-mean-squared error between the simulated data from [1]} and our theory is \(0.07\) .
